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Direction-dependent turning leads to anisotropic diffusion and persistence

Published online by Cambridge University Press:  23 June 2021

N. LOY
Affiliation:
Department of Mathematical Sciences “G.L. Lagrange”, Politecnico di Torino, C.so Duca degli Abruzzi, 24 Torino, Italy emails: nadia.loy@polito.it
T. HILLEN
Affiliation:
Department of Mathematical and Statistical Science, University of Alberta, Edmonton, AB, Canada email: thillen@ualberta.ca
K. J. PAINTER
Affiliation:
Interuniversity Department of Regional and Urban Studies and Planning, Politecnico di Torino, Viale Pier Andrea Mattioli, 39, Torino, Italy email: kevin.painter@polito.it

Abstract

Cells and organisms follow aligned structures in their environment, a process that can generate persistent migration paths. Kinetic transport equations are a popular modelling tool for describing biological movements at the mesoscopic level, yet their formulations usually assume a constant turning rate. Here we relax this simplification, extending to include a turning rate that varies according to the anisotropy of a heterogeneous environment. We extend known methods of parabolic and hyperbolic scaling and apply the results to cell movement on micropatterned domains. We show that inclusion of orientation dependence in the turning rate can lead to persistence of motion in an otherwise fully symmetric environment and generate enhanced diffusion in structured domains.

Type
Papers
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Adler, J. (1966) Chemotaxis in bacteria. Science 153(3737), 708–116.CrossRefGoogle Scholar
Alberts, B., Johnson, A. D., Lewis, J., Morgan, D., Raff, M., Roberts, K. & Walter, P. (2014) Molecular biology of the cell, Garland Sciences, New York.Google Scholar
Bisi, M., Carrillo, J. A. & Lods, B. (2008) Equilibrium solution to the inelastic Boltzmann equation driven by a particle bath. J. Stat. Phys. 133(5), 841870.CrossRefGoogle Scholar
Budrene, E. O. & Berg, H. C. (1991) Complex patterns formed by motile cells of escherichia coli. Nature 349(6310), 630633.CrossRefGoogle ScholarPubMed
Buttenschoen, A. & Hillen, T. (2021) Non-local Cell Adhesion Models: Symmetries and Bifurcations in 1-D, Springer, Heidelberg.CrossRefGoogle Scholar
Cercignani, C. (1987) The Boltzmann Equation and its Applications, Springer, New York.Google Scholar
Chalub, F. A. C. C., Markowich, P. A., Perthame, B. & Schmeiser, C. (2004) Kinetic models for chemotaxis and their drift-diffusion limits. Monatshefte fÜr Mathematik 142(1), 123141.CrossRefGoogle Scholar
Chapman, S. (1928) On the Brownian displacements and thermal diffusion of grains suspended in a non-uniform fluid. Proc. R. Soc. London A 119, 3454.Google Scholar
Chen, L., Painter, K. J., Surulescu, C. & Zhigun, A. (2020) Mathematical models for cell migration: a non-local perspective. Philos. Trans. R. Soc. B 375(1807), 20190379.CrossRefGoogle ScholarPubMed
Cho, E. & Kim, Y. J. (2013) Starvation driven diffusion as a survival strategy of biological organisms. Bull. Math. Biol. 75, 845870.CrossRefGoogle ScholarPubMed
Chung, J., Kim, Y. J., Kwong, O. & Yoon, C. W. (2020) Biological advection and cross diffusion with parameter regimes. AIMS Math. 4(6).CrossRefGoogle Scholar
Dallon, J. C., Sherratt, J. A. & Maini, P. K. (1999) Mathematical modelling of extracellular matrix dynamics using discrete cells: fiber orientation and tissue regeneration. J. Theor. Biol. 199(4), 449471.CrossRefGoogle ScholarPubMed
Degond, P., Goudon, T. & Poupaud, F. (2000) Diffusion limit for non homogeneous and non-micro-reversible processes. Indiana Univ. Math. J. 49(3), 11751198.Google Scholar
Dickinson, R. B. (2000) A generalized transport model for biased cell migration in an anisotropic environment. J. Math. Biol. 40(2), 97135.CrossRefGoogle Scholar
Dickinson, R. B. & Tranquillo, R. T. (1991) Stochastic model of biased cell migration based on binding fluctuations of adhesion receptors. 19, 563–600.Google Scholar
Doyle, A. D., Wang, F. W., Matsumoto, K. & Yamada, K. M. (2009) One-dimensional topography underlies three-dimensional fibrillar cell migration. J. Cell Biol. 184(4), 481–490.CrossRefGoogle Scholar
Dunn, G. A. & Heath, J. P. (1976) A new hypothesis of contact guidance in tissue cells. Exp. Cell Res. 101(1), 114.CrossRefGoogle ScholarPubMed
Engwer, C., Hillen, T., Knappitsch, M. & Surulescu, C. (2015) Glioma follow white matter tracts: a multiscale DTI-based model. J. Math. Biol. 71(3), 551582.CrossRefGoogle ScholarPubMed
Filbet, F. & Yang, C. (2010) Numerical simulation of a kinetic model for chemotaxis. Kinetic Related Models 3, B348B366.Google Scholar
Giese, A., Kluwe, L., Laube, B., Meissner, H., Berens, M. E. & Westphal, M. (1996) Migration of human glioma cells on myelin. Neurosurgery 38(4), 755764.CrossRefGoogle ScholarPubMed
Giese, A. & Westphal, M. (1996) Glioma invasion in the central nervous system. Neurosurgery 39(2), 235252.CrossRefGoogle ScholarPubMed
Goudon, T. & Mellet, A. Homogenization and diffusion asymptotics of the linear Boltzmann equation. ESAIM Control Optim. Calc. Var. 9, 371398 (2003).CrossRefGoogle Scholar
Hecht, I., Bar-El, Y., Balmer, F., Natan, S., Tsarfaty, I., Schweitzer, F. & Ben-Jacob, E. (2015) Tumor invasion optimization by mesenchymal-amoeboid heterogeneity. Sci. Rep. 5(10622).CrossRefGoogle Scholar
Hillen, T. (2006) M5 mesoscopic and macroscopic models for mesenchymal motion. J. Math. Biol. 53(4), 585616.CrossRefGoogle ScholarPubMed
Hillen, T., Murtha, A., Painter, K. J. & Swan, A. (2017) Moments of the von Mises and Fisher distributions and applications. Math. Biosci. Eng. 14(3), 673694.CrossRefGoogle ScholarPubMed
Hillen, T. & Othmer, H. G. (2000) The diffusion limit of transport equations derived from velocity-jump processes. SIAM J. Appl. Math. 61, 751775.Google Scholar
Hillen, T. & Painter, K. J. (2008) A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58(1), 183217.CrossRefGoogle ScholarPubMed
Hillen, T. & Painter, K. J. (2013) Transport and Anisotropic Diffusion Models for Movement in Oriented Habitats (pp. 177222), Vol. 2071, Springer-Verlag.Google Scholar
Lods, B. (2005) Semigroup generation properties of streaming operators with noncontractive boundary conditions. Math. Comput. Modell. 42, 14411462.CrossRefGoogle Scholar
Loy, N. & Preziosi, L. (2019) Kinetic models with non-local sensing determining cell polarization and speed according to independent cues. J. Math. Biol., 1–49.Google Scholar
Lutscher, F. & Hillen, T. (2021) Homogenization of correlated random walks in heterogeneous landscapes. AIMS Math.Google Scholar
McDougall, S., Dallon, J., Sherratt, J. & Maini, P. (2006) Fibroblast migration and collagen deposition during dermal wound healing: mathematical modelling and clinical implications. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 364(1843), 13851405.CrossRefGoogle ScholarPubMed
Othmer, H. & Stevens, A. (2001) Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 10441081.Google Scholar
Othmer, H. G., Dunbar, S. R. & Alt, W. (1988) Models of dispersal in biological systems. J. Math. Biol. 26(3), 263298.CrossRefGoogle ScholarPubMed
Othmer, H. G. & Hillen, T. (2002) The diffusion limit of transport equations II: chemotaxis equations. SIAM J. Appl. Math. 62, 12221250.CrossRefGoogle Scholar
Painter, K. J. (2009) Modelling cell migration strategies in the extracellular matrix. J. Math. Biol. 58(4), 511543.CrossRefGoogle ScholarPubMed
Painter, K. J. & Hillen, T. (2013) Mathematical modelling of glioma growth: the use of diffusion tensor imaging (DTI) data to predict the anisotropic pathways of cancer invasion. J. Theor. Biol. 323, 2539.CrossRefGoogle ScholarPubMed
Painter, K. J. & Hillen, T. (2019) From random walks to fully anisotropic diffusion models for cell and animal movement. In Modeling and Simulation in Science, Engineering and Technology, Birkhauser.Google Scholar
Palecek, S. P., Loftus, J. C., Ginsberg, M. H., Lauffenburger, D. A. & Horwitz, A. F. (1997) Integrin-ligand binding properties govern cell migration speed through cell-substratum adhesiveness. Nature 385(6616), 537–540.Google Scholar
Petterson, R. (1983) Existence theorems for the linear, space-inhomogeneous transport equation. IMA J. Appl. Math. 30(1), 81105.CrossRefGoogle Scholar
Pettersson, R. (2004) On solutions to the linear Boltzmann equation for granular gases. Transp. Theory Stat. Phys. 33(5–7), 527543.CrossRefGoogle Scholar
Plaza, R. G. (2019) Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process. J. Math. Biol. 78, 16811711.CrossRefGoogle ScholarPubMed
Ray, A., Morford, R. K., Ghaderi, N., Odde, D. J. & Provenzano, P. P. (2018) Dynamics of 3D carcinoma cell invasion into aligned collagen. Integr. Biol. 10(2), 100112.CrossRefGoogle ScholarPubMed
Ray, A., Slama, Z., Morford, R., Madden, S. & Provenzano, P. (2017) Enhanced directional migration of cancer stem cells in 3D aligned collagen matrices. Biophys. J. 112, 10231036.CrossRefGoogle ScholarPubMed
Riching, K. M., Cox, B. L., Salick, M. R., Pehlke, C., Riching, A. S., Ponik, S. M., Bass, B. R., Crone, W. C., Jiang, Y., Weaver, A. M., Eliceiri, K. W. & Keely, P. J. (2014) 3d collagen alignment limits protrusions to enhance breast cancer cell persistence. Biophys. J. 107(11), 25462558.CrossRefGoogle ScholarPubMed
SchlÜter, D. K., Ramis-Conde, I. & Chaplain, M. A. J. (2012) Computational modeling of single-cell migration: the leading role of extracellular matrix fibers. Biophys. J. 103(6), 11411151.CrossRefGoogle ScholarPubMed
Scianna, M. & Preziosi, L. (2013) Modeling the influence of nucleus elasticity on cell invasion in fiber networks and microchannels. J. Theor. Biol. 317, 394406.CrossRefGoogle ScholarPubMed
Scianna, M. & Preziosi, L. (2014) A cellular Potts model for the MMP-dependent and-independent cancer cell migration in matrix microtracks of different dimensions. Comput. Mech. 53, 485497.CrossRefGoogle Scholar
Scianna, M., Preziosi, L. & Wolf, K. (2013) A Cellular Potts Model simulating cell migration on and in matrix environments. Math. Biosci. Eng. 10, 235261.Google ScholarPubMed
Swan, A. (2016) An Anisotropic Diffusion Model for Brain Tumour Spread. PhD thesis, University of Alberta.Google Scholar
Swan, A., Hillen, T., Bowman, J. & Murtha, A. (2017) An anisotropic model for glioma spread. Bull. Math. Biol. 80(5), 12591291.CrossRefGoogle ScholarPubMed
Talkenberger, K., Cavalcanti-Adam, E. A., Voss-Böhme, A. & Deutsch, A. (2017) Amoeboid-mesenchymal migration plasticity promotes invasion only in complex heterogeneous microenvironments. Sci. Rep. 7(9237).CrossRefGoogle Scholar
Te Boekhorst, V., Preziosi, L. & Friedl, P. (2016) Plasticity of cell migration in vivo and in silico. Ann. Rev. Cell Develop. Biol. 32, 491526.CrossRefGoogle ScholarPubMed
ThÉry, M. (2010) Micropatterning as a tool to decipher cell morphogenesis and functions. J. Cell Sci. 123(24), 42014213.CrossRefGoogle ScholarPubMed
Wereide, M. Th. (1914) La diffusion d’une solution dont la concentration et la temperature sont variables. Annales de Physique 2, 6783.CrossRefGoogle Scholar
Wolf, K., Mazo, I., Leung, I., Engelke, K., von Andria, U., Deryngina, E. I., Stron gin, A. Y., Brocker, E. B. & Friedl, P. (2003) Compensation mechanism in tumor cell migration: mesenchymal-amoeboid transition after blocking of pericellular proteolysis. J. Cell Biol. 160, 267–277.CrossRefGoogle Scholar